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Three-dimensional finite-different time-domain (3D FDTD) simulation of photonic crystal devices often demands large amount of computational resources. In many cases it is unlikely to carry out the task on a serial computer. We have therefore parallelized a 3D FDTD code using MPI. Initially we used a one-dimensional topology so that the computational domain was divided into slices perpendicular to the direction of the power flow. Even though the speed-up of this implementation left considerable room for improvement, we were nevertheless able to solve largescale and long-running problems. Two such cases were studied: the power transmission in a two-dimensional photonic crystal waveguide in a multilayered structure, and the power coupling from a wire waveguide to a photonic crystal slab. In the first case, a power dip due to TE/TM modes conversion is observed and in the second case, the structure is optimized to improve the coupling. We have also recently completed a full three-dimensional topology parallelization of the FDTD code.
The Doi kinetic theory for homogeneous flows of rodlike liquid crystalline polymers (LCPs) is extended to inhomogeneous flows through introducing a nonlocal intermolecular potential. An extra term in the form of an elastic body force comes out as a result of this extension. Systematic asympototic analysis in the small Deborah number limit is carried out, and the classical Ericksen- Leslie equations are derived in this limit. The Leslie coefficients are derived in terms of molecular parameters, and the Ericksen stress emerges from the body force.
In this paper, the jump conditions for the normal derivative of the pressure have been derived for two-phase Stokes (and Navier-Stokes) equations with discontinuous viscosity and singular sources in two and three dimensions. While different jump conditions for the pressure and the velocity can be found in the literature, the jump condition of the normal derivative of the pressure is new. The derivation is based on the idea of the immersed interface method [9, 8] that uses a fixed local coordinate system and the balance of forces along the interface that separates the two phases. The derivation process also provides a way to compute the jump conditions. The jump conditions for the pressure and the velocity are useful in developing accurate numerical methods for two-phase Stokes equations and Navier-Stokes equations.